Stericated 8-simplexes

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8-simplex t0.svg
8-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t04.svg
Stericated 8-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t15.svg
Bistericated 8-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t014.svg
Steri-truncated 8-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t125.svg
Bisteri-truncated 8-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t014.svg
Steri-cantellated 8-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t125.svg
Bisteri-cantellated 8-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t0124.svg
Stericanti-truncated 8-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t1235.svg
Bistericanti-truncated 8-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t034.svg
Steri-runcinated 8-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t145.svg
Bisteri-runcinated 8-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t0134.svg
Steriruncitruncated 8-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t1245.svg
Bisterirun-citruncated 8-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t0134.svg
Sterirunci-cantellated 8-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t1245.svg
Bisterirunci-cantellated 8-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t01234.svg
Steriruncicanti-truncated 8-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
8-simplex t12345.svg
Bisteriruncicanti-truncated 8-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Orthogonal projections in A8 Coxeter plane

In eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations (sterication) of the regular 8-simplex. There are 16 unique sterications for the 8-simplex including permutations of truncation, cantellation, and runcination.

Stericated 8-simplex[edit]

Stericated 8-simplex
Type uniform 8-polytope
Schläfli symbol t0,4{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 6300
Vertices 630
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

Coordinates[edit]

The Cartesian coordinates of the vertices of the stericated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,1,2). This construction is based on facets of the stericated 9-orthoplex.

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t04.svg 8-simplex t04 A7.svg 8-simplex t04 A6.svg 8-simplex t04 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t04 A4.svg 8-simplex t04 A3.svg 8-simplex t04 A2.svg
Dihedral symmetry [5] [4] [3]

Bistericated 8-simplex[edit]

bistericated 8-simplex
Type uniform 8-polytope
Schläfli symbol t1,5{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 12600
Vertices 1260
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

Coordinates[edit]

The Cartesian coordinates of the vertices of the bistericated 8-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,1,2,2). This construction is based on facets of the bistericated 9-orthoplex.

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t15.svg 8-simplex t15 A7.svg 8-simplex t15 A6.svg 8-simplex t15 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t15 A4.svg 8-simplex t15 A3.svg 8-simplex t15 A2.svg
Dihedral symmetry [5] [4] [3]

Steritruncated 8-simplex[edit]

Steritruncated 8-simplex
Type uniform 8-polytope
Schläfli symbol t0,1,4{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t014.svg 8-simplex t014 A7.svg 8-simplex t014 A6.svg 8-simplex t014 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t014 A4.svg 8-simplex t014 A3.svg 8-simplex t014 A2.svg
Dihedral symmetry [5] [4] [3]

Bisteritruncated 8-simplex[edit]

Bisteritruncated 8-simplex
Type uniform 8-polytope
Schläfli symbol t1,2,5{3,3,3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t125.svg 8-simplex t125 A7.svg 8-simplex t125 A6.svg 8-simplex t125 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t125 A4.svg 8-simplex t125 A3.svg 8-simplex t125 A2.svg
Dihedral symmetry [5] [4] [3]

Stericantellated 8-simplex[edit]

CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t024.svg 8-simplex t024 A7.svg 8-simplex t024 A6.svg 8-simplex t024 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t024 A4.svg 8-simplex t024 A3.svg 8-simplex t024 A2.svg
Dihedral symmetry [5] [4] [3]

Bistericantellated 8-simplex[edit]

CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t135.svg 8-simplex t135 A7.svg 8-simplex t135 A6.svg 8-simplex t135 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t135 A4.svg 8-simplex t135 A3.svg 8-simplex t135 A2.svg
Dihedral symmetry [5] [4] [3]

Stericantitruncated 8-simplex[edit]

CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t0124.svg 8-simplex t0124 A7.svg 8-simplex t0124 A6.svg 8-simplex t0124 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t0124 A4.svg 8-simplex t0124 A3.svg 8-simplex t0124 A2.svg
Dihedral symmetry [5] [4] [3]

Bistericantitruncated 8-simplex[edit]

CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t1235.svg 8-simplex t1235 A7.svg 8-simplex t1235 A6.svg 8-simplex t1235 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t1235 A4.svg 8-simplex t1235 A3.svg 8-simplex t1235 A2.svg
Dihedral symmetry [5] [4] [3]

Steriruncinated 8-simplex[edit]

CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t034.svg 8-simplex t034 A7.svg 8-simplex t034 A6.svg 8-simplex t034 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t034 A4.svg 8-simplex t034 A3.svg 8-simplex t034 A2.svg
Dihedral symmetry [5] [4] [3]

Bisteriruncinated 8-simplex[edit]

CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t145.svg 8-simplex t145 A7.svg 8-simplex t145 A6.svg 8-simplex t145 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t145 A4.svg 8-simplex t145 A3.svg 8-simplex t145 A2.svg
Dihedral symmetry [5] [4] [3]

Steriruncitruncated 8-simplex[edit]

CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t0134.svg 8-simplex t0134 A7.svg 8-simplex family graph.svg 8-simplex t0134 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t0134 A4.svg 8-simplex t0134 A3.svg 8-simplex t0134 A2.svg
Dihedral symmetry [5] [4] [3]

Bisteriruncitruncated 8-simplex[edit]

CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t1245.svg 8-simplex t1245 A7.svg 8-simplex t1245 A6.svg 8-simplex t1245 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t1245 A4.svg 8-simplex t1245 A3.svg 8-simplex t1245 A2.svg
Dihedral symmetry [5] [4] [3]

Steriruncicantellated 8-simplex[edit]

CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t0234.svg 8-simplex t0234 A7.svg 8-simplex t0234 A6.svg 8-simplex t0234 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t0234 A4.svg 8-simplex t0234 A3.svg 8-simplex t0234 A2.svg
Dihedral symmetry [5] [4] [3]

Bisteriruncicantellated 8-simplex[edit]

CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t1345.svg 8-simplex t1345 A7.svg 8-simplex t1345 A6.svg 8-simplex t1345 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t1345 A4.svg 8-simplex t1345 A3.svg 8-simplex t1345 A2.svg
Dihedral symmetry [5] [4] [3]

Steriruncicantitruncated 8-simplex[edit]

CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t01234.svg 8-simplex t01234 A7.svg 8-simplex t01234 A6.svg 8-simplex t01234 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t01234 A4.svg 8-simplex t01234 A3.svg 8-simplex t01234 A2.svg
Dihedral symmetry [5] [4] [3]

Bisteriruncicantitruncated 8-simplex[edit]

CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Images[edit]

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph 8-simplex t12345.svg 8-simplex t12345 A7.svg 8-simplex t12345 A6.svg 8-simplex t12345 A5.svg
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 8-simplex t12345 A4.svg 8-simplex t12345 A3.svg 8-simplex t12345 A2.svg
Dihedral symmetry [5] [4] [3]

Related polytopes[edit]

This polytope is one of 135 uniform 8-polytopes with A8 symmetry.

Notes[edit]

References[edit]

  • H.S.M. Coxeter:
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
    • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
      • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
      • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
  • Klitzing, Richard. "8D uniform polytopes (polyzetta)".  x3o3o3o3x3o3o3o, o3x3o3o3o3x3o3o

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds